I have an ATI something (can't remember the exact model). Its got DVI x2 and HDMI.
At the moment I have a couple of old 1440x900 monitors connected to DVI and my 1080p TV connected by HDMI. I have a shortcut to a script that turns off the primary monitor, turns on the HDMI output to the TV, switched the audio device to use HDMI out and opens Steam in big picture.
I'm going to get a 1080p monitor so don't need to worry about changing primary monitor if they both run the same resolution (monitor and TV), I could just mirror the HDMI out across both screens. I could have course just keep the script I have too, as I'd need to still swap the audio.
My card won't do 3 outputs at once, it can only have 2 of the 3 active at any one time, hence the switching.